2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
8 from itertools
import izip
, repeat
10 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
11 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
12 from sage
.combinat
.free_module
import CombinatorialFreeModule
13 from sage
.matrix
.constructor
import matrix
14 from sage
.matrix
.matrix_space
import MatrixSpace
15 from sage
.misc
.cachefunc
import cached_method
16 from sage
.misc
.prandom
import choice
17 from sage
.misc
.table
import table
18 from sage
.modules
.free_module
import FreeModule
, VectorSpace
19 from sage
.rings
.integer_ring
import ZZ
20 from sage
.rings
.number_field
.number_field
import NumberField
, QuadraticField
21 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
22 from sage
.rings
.rational_field
import QQ
23 from sage
.rings
.real_lazy
import CLF
, RLF
24 from sage
.structure
.element
import is_Matrix
26 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
27 from mjo
.eja
.eja_utils
import _mat2vec
29 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
30 # This is an ugly hack needed to prevent the category framework
31 # from implementing a coercion from our base ring (e.g. the
32 # rationals) into the algebra. First of all -- such a coercion is
33 # nonsense to begin with. But more importantly, it tries to do so
34 # in the category of rings, and since our algebras aren't
35 # associative they generally won't be rings.
36 _no_generic_basering_coercion
= True
48 sage: from mjo.eja.eja_algebra import random_eja
52 By definition, Jordan multiplication commutes::
54 sage: set_random_seed()
55 sage: J = random_eja()
56 sage: x,y = J.random_elements(2)
62 self
._natural
_basis
= natural_basis
65 category
= MagmaticAlgebras(field
).FiniteDimensional()
66 category
= category
.WithBasis().Unital()
68 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
70 range(len(mult_table
)),
73 self
.print_options(bracket
='')
75 # The multiplication table we're given is necessarily in terms
76 # of vectors, because we don't have an algebra yet for
77 # anything to be an element of. However, it's faster in the
78 # long run to have the multiplication table be in terms of
79 # algebra elements. We do this after calling the superclass
80 # constructor so that from_vector() knows what to do.
81 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
82 for ls
in mult_table
]
85 def _element_constructor_(self
, elt
):
87 Construct an element of this algebra from its natural
90 This gets called only after the parent element _call_ method
91 fails to find a coercion for the argument.
95 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
96 ....: RealCartesianProductEJA,
97 ....: RealSymmetricEJA)
101 The identity in `S^n` is converted to the identity in the EJA::
103 sage: J = RealSymmetricEJA(3)
104 sage: I = matrix.identity(QQ,3)
105 sage: J(I) == J.one()
108 This skew-symmetric matrix can't be represented in the EJA::
110 sage: J = RealSymmetricEJA(3)
111 sage: A = matrix(QQ,3, lambda i,j: i-j)
113 Traceback (most recent call last):
115 ArithmeticError: vector is not in free module
119 Ensure that we can convert any element of the two non-matrix
120 simple algebras (whose natural representations are their usual
121 vector representations) back and forth faithfully::
123 sage: set_random_seed()
124 sage: J = RealCartesianProductEJA.random_instance()
125 sage: x = J.random_element()
126 sage: J(x.to_vector().column()) == x
128 sage: J = JordanSpinEJA.random_instance()
129 sage: x = J.random_element()
130 sage: J(x.to_vector().column()) == x
135 # The superclass implementation of random_element()
136 # needs to be able to coerce "0" into the algebra.
139 natural_basis
= self
.natural_basis()
140 basis_space
= natural_basis
[0].matrix_space()
141 if elt
not in basis_space
:
142 raise ValueError("not a naturally-represented algebra element")
144 # Thanks for nothing! Matrix spaces aren't vector spaces in
145 # Sage, so we have to figure out its natural-basis coordinates
146 # ourselves. We use the basis space's ring instead of the
147 # element's ring because the basis space might be an algebraic
148 # closure whereas the base ring of the 3-by-3 identity matrix
149 # could be QQ instead of QQbar.
150 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
151 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
152 coords
= W
.coordinate_vector(_mat2vec(elt
))
153 return self
.from_vector(coords
)
157 def _max_test_case_size():
159 Return an integer "size" that is an upper bound on the size of
160 this algebra when it is used in a random test
161 case. Unfortunately, the term "size" is quite vague -- when
162 dealing with `R^n` under either the Hadamard or Jordan spin
163 product, the "size" refers to the dimension `n`. When dealing
164 with a matrix algebra (real symmetric or complex/quaternion
165 Hermitian), it refers to the size of the matrix, which is
166 far less than the dimension of the underlying vector space.
168 We default to five in this class, which is safe in `R^n`. The
169 matrix algebra subclasses (or any class where the "size" is
170 interpreted to be far less than the dimension) should override
171 with a smaller number.
178 Return a string representation of ``self``.
182 sage: from mjo.eja.eja_algebra import JordanSpinEJA
186 Ensure that it says what we think it says::
188 sage: JordanSpinEJA(2, field=QQ)
189 Euclidean Jordan algebra of dimension 2 over Rational Field
190 sage: JordanSpinEJA(3, field=RDF)
191 Euclidean Jordan algebra of dimension 3 over Real Double Field
194 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
195 return fmt
.format(self
.dimension(), self
.base_ring())
197 def product_on_basis(self
, i
, j
):
198 return self
._multiplication
_table
[i
][j
]
200 def _a_regular_element(self
):
202 Guess a regular element. Needed to compute the basis for our
203 characteristic polynomial coefficients.
207 sage: from mjo.eja.eja_algebra import random_eja
211 Ensure that this hacky method succeeds for every algebra that we
212 know how to construct::
214 sage: set_random_seed()
215 sage: J = random_eja()
216 sage: J._a_regular_element().is_regular()
221 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
222 if not z
.is_regular():
223 raise ValueError("don't know a regular element")
228 def _charpoly_basis_space(self
):
230 Return the vector space spanned by the basis used in our
231 characteristic polynomial coefficients. This is used not only to
232 compute those coefficients, but also any time we need to
233 evaluate the coefficients (like when we compute the trace or
236 z
= self
._a
_regular
_element
()
237 # Don't use the parent vector space directly here in case this
238 # happens to be a subalgebra. In that case, we would be e.g.
239 # two-dimensional but span_of_basis() would expect three
241 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
242 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
243 V1
= V
.span_of_basis( basis
)
244 b
= (V1
.basis() + V1
.complement().basis())
245 return V
.span_of_basis(b
)
250 def _charpoly_coeff(self
, i
):
252 Return the coefficient polynomial "a_{i}" of this algebra's
253 general characteristic polynomial.
255 Having this be a separate cached method lets us compute and
256 store the trace/determinant (a_{r-1} and a_{0} respectively)
257 separate from the entire characteristic polynomial.
259 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
260 R
= A_of_x
.base_ring()
262 # Guaranteed by theory
265 # Danger: the in-place modification is done for performance
266 # reasons (reconstructing a matrix with huge polynomial
267 # entries is slow), but I don't know how cached_method works,
268 # so it's highly possible that we're modifying some global
269 # list variable by reference, here. In other words, you
270 # probably shouldn't call this method twice on the same
271 # algebra, at the same time, in two threads
272 Ai_orig
= A_of_x
.column(i
)
273 A_of_x
.set_column(i
,xr
)
274 numerator
= A_of_x
.det()
275 A_of_x
.set_column(i
,Ai_orig
)
277 # We're relying on the theory here to ensure that each a_i is
278 # indeed back in R, and the added negative signs are to make
279 # the whole charpoly expression sum to zero.
280 return R(-numerator
/detA
)
284 def _charpoly_matrix_system(self
):
286 Compute the matrix whose entries A_ij are polynomials in
287 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
288 corresponding to `x^r` and the determinent of the matrix A =
289 [A_ij]. In other words, all of the fixed (cachable) data needed
290 to compute the coefficients of the characteristic polynomial.
295 # Turn my vector space into a module so that "vectors" can
296 # have multivatiate polynomial entries.
297 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
298 R
= PolynomialRing(self
.base_ring(), names
)
300 # Using change_ring() on the parent's vector space doesn't work
301 # here because, in a subalgebra, that vector space has a basis
302 # and change_ring() tries to bring the basis along with it. And
303 # that doesn't work unless the new ring is a PID, which it usually
307 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
311 # And figure out the "left multiplication by x" matrix in
314 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
315 for i
in range(n
) ] # don't recompute these!
317 ek
= self
.monomial(k
).to_vector()
319 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
320 for i
in range(n
) ) )
321 Lx
= matrix
.column(R
, lmbx_cols
)
323 # Now we can compute powers of x "symbolically"
324 x_powers
= [self
.one().to_vector(), x
]
325 for d
in range(2, r
+1):
326 x_powers
.append( Lx
*(x_powers
[-1]) )
328 idmat
= matrix
.identity(R
, n
)
330 W
= self
._charpoly
_basis
_space
()
331 W
= W
.change_ring(R
.fraction_field())
333 # Starting with the standard coordinates x = (X1,X2,...,Xn)
334 # and then converting the entries to W-coordinates allows us
335 # to pass in the standard coordinates to the charpoly and get
336 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
339 # W.coordinates(x^2) eval'd at (standard z-coords)
343 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
345 # We want the middle equivalent thing in our matrix, but use
346 # the first equivalent thing instead so that we can pass in
347 # standard coordinates.
348 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
349 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
350 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
351 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
355 def characteristic_polynomial(self
):
357 Return a characteristic polynomial that works for all elements
360 The resulting polynomial has `n+1` variables, where `n` is the
361 dimension of this algebra. The first `n` variables correspond to
362 the coordinates of an algebra element: when evaluated at the
363 coordinates of an algebra element with respect to a certain
364 basis, the result is a univariate polynomial (in the one
365 remaining variable ``t``), namely the characteristic polynomial
370 sage: from mjo.eja.eja_algebra import JordanSpinEJA
374 The characteristic polynomial in the spin algebra is given in
375 Alizadeh, Example 11.11::
377 sage: J = JordanSpinEJA(3)
378 sage: p = J.characteristic_polynomial(); p
379 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
380 sage: xvec = J.one().to_vector()
388 # The list of coefficient polynomials a_1, a_2, ..., a_n.
389 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
391 # We go to a bit of trouble here to reorder the
392 # indeterminates, so that it's easier to evaluate the
393 # characteristic polynomial at x's coordinates and get back
394 # something in terms of t, which is what we want.
396 S
= PolynomialRing(self
.base_ring(),'t')
398 S
= PolynomialRing(S
, R
.variable_names())
401 # Note: all entries past the rth should be zero. The
402 # coefficient of the highest power (x^r) is 1, but it doesn't
403 # appear in the solution vector which contains coefficients
404 # for the other powers (to make them sum to x^r).
406 a
[r
] = 1 # corresponds to x^r
408 # When the rank is equal to the dimension, trying to
409 # assign a[r] goes out-of-bounds.
410 a
.append(1) # corresponds to x^r
412 return sum( a
[k
]*(t
**k
) for k
in xrange(len(a
)) )
415 def inner_product(self
, x
, y
):
417 The inner product associated with this Euclidean Jordan algebra.
419 Defaults to the trace inner product, but can be overridden by
420 subclasses if they are sure that the necessary properties are
425 sage: from mjo.eja.eja_algebra import random_eja
429 Our inner product satisfies the Jordan axiom, which is also
430 referred to as "associativity" for a symmetric bilinear form::
432 sage: set_random_seed()
433 sage: J = random_eja()
434 sage: x,y,z = J.random_elements(3)
435 sage: (x*y).inner_product(z) == y.inner_product(x*z)
439 X
= x
.natural_representation()
440 Y
= y
.natural_representation()
441 return self
.natural_inner_product(X
,Y
)
444 def is_trivial(self
):
446 Return whether or not this algebra is trivial.
448 A trivial algebra contains only the zero element.
452 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
456 sage: J = ComplexHermitianEJA(3)
459 sage: A = J.zero().subalgebra_generated_by()
464 return self
.dimension() == 0
467 def multiplication_table(self
):
469 Return a visual representation of this algebra's multiplication
470 table (on basis elements).
474 sage: from mjo.eja.eja_algebra import JordanSpinEJA
478 sage: J = JordanSpinEJA(4)
479 sage: J.multiplication_table()
480 +----++----+----+----+----+
481 | * || e0 | e1 | e2 | e3 |
482 +====++====+====+====+====+
483 | e0 || e0 | e1 | e2 | e3 |
484 +----++----+----+----+----+
485 | e1 || e1 | e0 | 0 | 0 |
486 +----++----+----+----+----+
487 | e2 || e2 | 0 | e0 | 0 |
488 +----++----+----+----+----+
489 | e3 || e3 | 0 | 0 | e0 |
490 +----++----+----+----+----+
493 M
= list(self
._multiplication
_table
) # copy
494 for i
in xrange(len(M
)):
495 # M had better be "square"
496 M
[i
] = [self
.monomial(i
)] + M
[i
]
497 M
= [["*"] + list(self
.gens())] + M
498 return table(M
, header_row
=True, header_column
=True, frame
=True)
501 def natural_basis(self
):
503 Return a more-natural representation of this algebra's basis.
505 Every finite-dimensional Euclidean Jordan Algebra is a direct
506 sum of five simple algebras, four of which comprise Hermitian
507 matrices. This method returns the original "natural" basis
508 for our underlying vector space. (Typically, the natural basis
509 is used to construct the multiplication table in the first place.)
511 Note that this will always return a matrix. The standard basis
512 in `R^n` will be returned as `n`-by-`1` column matrices.
516 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
517 ....: RealSymmetricEJA)
521 sage: J = RealSymmetricEJA(2)
523 Finite family {0: e0, 1: e1, 2: e2}
524 sage: J.natural_basis()
526 [1 0] [ 0 1/2*sqrt2] [0 0]
527 [0 0], [1/2*sqrt2 0], [0 1]
532 sage: J = JordanSpinEJA(2)
534 Finite family {0: e0, 1: e1}
535 sage: J.natural_basis()
542 if self
._natural
_basis
is None:
543 M
= self
.natural_basis_space()
544 return tuple( M(b
.to_vector()) for b
in self
.basis() )
546 return self
._natural
_basis
549 def natural_basis_space(self
):
551 Return the matrix space in which this algebra's natural basis
554 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
555 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
557 return self
._natural
_basis
[0].matrix_space()
561 def natural_inner_product(X
,Y
):
563 Compute the inner product of two naturally-represented elements.
565 For example in the real symmetric matrix EJA, this will compute
566 the trace inner-product of two n-by-n symmetric matrices. The
567 default should work for the real cartesian product EJA, the
568 Jordan spin EJA, and the real symmetric matrices. The others
569 will have to be overridden.
571 return (X
.conjugate_transpose()*Y
).trace()
577 Return the unit element of this algebra.
581 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
586 sage: J = RealCartesianProductEJA(5)
588 e0 + e1 + e2 + e3 + e4
592 The identity element acts like the identity::
594 sage: set_random_seed()
595 sage: J = random_eja()
596 sage: x = J.random_element()
597 sage: J.one()*x == x and x*J.one() == x
600 The matrix of the unit element's operator is the identity::
602 sage: set_random_seed()
603 sage: J = random_eja()
604 sage: actual = J.one().operator().matrix()
605 sage: expected = matrix.identity(J.base_ring(), J.dimension())
606 sage: actual == expected
610 # We can brute-force compute the matrices of the operators
611 # that correspond to the basis elements of this algebra.
612 # If some linear combination of those basis elements is the
613 # algebra identity, then the same linear combination of
614 # their matrices has to be the identity matrix.
616 # Of course, matrices aren't vectors in sage, so we have to
617 # appeal to the "long vectors" isometry.
618 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
620 # Now we use basis linear algebra to find the coefficients,
621 # of the matrices-as-vectors-linear-combination, which should
622 # work for the original algebra basis too.
623 A
= matrix
.column(self
.base_ring(), oper_vecs
)
625 # We used the isometry on the left-hand side already, but we
626 # still need to do it for the right-hand side. Recall that we
627 # wanted something that summed to the identity matrix.
628 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
630 # Now if there's an identity element in the algebra, this should work.
631 coeffs
= A
.solve_right(b
)
632 return self
.linear_combination(zip(self
.gens(), coeffs
))
635 def random_element(self
):
636 # Temporary workaround for https://trac.sagemath.org/ticket/28327
637 if self
.is_trivial():
640 s
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
641 return s
.random_element()
643 def random_elements(self
, count
):
645 Return ``count`` random elements as a tuple.
649 sage: from mjo.eja.eja_algebra import JordanSpinEJA
653 sage: J = JordanSpinEJA(3)
654 sage: x,y,z = J.random_elements(3)
655 sage: all( [ x in J, y in J, z in J ])
657 sage: len( J.random_elements(10) ) == 10
661 return tuple( self
.random_element() for idx
in xrange(count
) )
664 def random_instance(cls
, field
=QQ
, **kwargs
):
666 Return a random instance of this type of algebra.
668 In subclasses for algebras that we know how to construct, this
669 is a shortcut for constructing test cases and examples.
671 if cls
is FiniteDimensionalEuclideanJordanAlgebra
:
672 # Red flag! But in theory we could do this I guess. The
673 # only finite-dimensional exceptional EJA is the
674 # octononions. So, we could just create an EJA from an
675 # associative matrix algebra (generated by a subset of
676 # elements) with the symmetric product. Or, we could punt
677 # to random_eja() here, override it in our subclasses, and
678 # not worry about it.
679 raise NotImplementedError
681 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
682 return cls(n
, field
, **kwargs
)
687 Return the rank of this EJA.
691 The author knows of no algorithm to compute the rank of an EJA
692 where only the multiplication table is known. In lieu of one, we
693 require the rank to be specified when the algebra is created,
694 and simply pass along that number here.
698 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
699 ....: RealSymmetricEJA,
700 ....: ComplexHermitianEJA,
701 ....: QuaternionHermitianEJA,
706 The rank of the Jordan spin algebra is always two::
708 sage: JordanSpinEJA(2).rank()
710 sage: JordanSpinEJA(3).rank()
712 sage: JordanSpinEJA(4).rank()
715 The rank of the `n`-by-`n` Hermitian real, complex, or
716 quaternion matrices is `n`::
718 sage: RealSymmetricEJA(4).rank()
720 sage: ComplexHermitianEJA(3).rank()
722 sage: QuaternionHermitianEJA(2).rank()
727 Ensure that every EJA that we know how to construct has a
728 positive integer rank::
730 sage: set_random_seed()
731 sage: r = random_eja().rank()
732 sage: r in ZZ and r > 0
739 def vector_space(self
):
741 Return the vector space that underlies this algebra.
745 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
749 sage: J = RealSymmetricEJA(2)
750 sage: J.vector_space()
751 Vector space of dimension 3 over...
754 return self
.zero().to_vector().parent().ambient_vector_space()
757 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
760 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
762 Return the Euclidean Jordan Algebra corresponding to the set
763 `R^n` under the Hadamard product.
765 Note: this is nothing more than the Cartesian product of ``n``
766 copies of the spin algebra. Once Cartesian product algebras
767 are implemented, this can go.
771 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
775 This multiplication table can be verified by hand::
777 sage: J = RealCartesianProductEJA(3)
778 sage: e0,e1,e2 = J.gens()
794 We can change the generator prefix::
796 sage: RealCartesianProductEJA(3, prefix='r').gens()
800 def __init__(self
, n
, field
=QQ
, **kwargs
):
801 V
= VectorSpace(field
, n
)
802 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in xrange(n
) ]
805 fdeja
= super(RealCartesianProductEJA
, self
)
806 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
808 def inner_product(self
, x
, y
):
810 Faster to reimplement than to use natural representations.
814 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
818 Ensure that this is the usual inner product for the algebras
821 sage: set_random_seed()
822 sage: J = RealCartesianProductEJA.random_instance()
823 sage: x,y = J.random_elements(2)
824 sage: X = x.natural_representation()
825 sage: Y = y.natural_representation()
826 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
830 return x
.to_vector().inner_product(y
.to_vector())
835 Return a "random" finite-dimensional Euclidean Jordan Algebra.
839 For now, we choose a random natural number ``n`` (greater than zero)
840 and then give you back one of the following:
842 * The cartesian product of the rational numbers ``n`` times; this is
843 ``QQ^n`` with the Hadamard product.
845 * The Jordan spin algebra on ``QQ^n``.
847 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
850 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
851 in the space of ``2n``-by-``2n`` real symmetric matrices.
853 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
854 in the space of ``4n``-by-``4n`` real symmetric matrices.
856 Later this might be extended to return Cartesian products of the
861 sage: from mjo.eja.eja_algebra import random_eja
866 Euclidean Jordan algebra of dimension...
869 classname
= choice([RealCartesianProductEJA
,
873 QuaternionHermitianEJA
])
874 return classname
.random_instance()
881 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
883 def _max_test_case_size():
884 # Play it safe, since this will be squared and the underlying
885 # field can have dimension 4 (quaternions) too.
889 def _denormalized_basis(cls
, n
, field
):
890 raise NotImplementedError
892 def __init__(self
, n
, field
=QQ
, normalize_basis
=True, **kwargs
):
893 S
= self
._denormalized
_basis
(n
, field
)
895 # Used in this class's fast _charpoly_coeff() override.
896 self
._basis
_normalizers
= None
898 if n
> 1 and normalize_basis
:
899 # We'll need sqrt(2) to normalize the basis, and this
900 # winds up in the multiplication table, so the whole
901 # algebra needs to be over the field extension.
902 R
= PolynomialRing(field
, 'z')
905 if p
.is_irreducible():
906 field
= NumberField(p
, 'sqrt2', embedding
=RLF(2).sqrt())
907 S
= [ s
.change_ring(field
) for s
in S
]
908 self
._basis
_normalizers
= tuple(
909 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in S
)
910 S
= tuple( s
*c
for (s
,c
) in zip(S
,self
._basis
_normalizers
) )
912 Qs
= self
.multiplication_table_from_matrix_basis(S
)
914 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
915 return fdeja
.__init
__(field
,
923 def _charpoly_coeff(self
, i
):
925 Override the parent method with something that tries to compute
926 over a faster (non-extension) field.
928 if self
._basis
_normalizers
is None:
929 # We didn't normalize, so assume that the basis we started
930 # with had entries in a nice field.
931 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coeff
(i
)
933 # If we didn't unembed first, this number would be wrong
934 # by a power-of-two factor for complex/quaternion matrices.
935 n
= self
.real_unembed(self
.natural_basis_space().zero()).nrows()
936 field
= self
.base_ring().base_ring() # yeeeeaaaahhh
937 J
= self
.__class
__(n
, field
, False)
938 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
939 p
= J
._charpoly
_coeff
(i
)
940 # p might be missing some vars, have to substitute "optionally"
941 pairs
= izip(x
.base_ring().gens(), self
._basis
_normalizers
)
942 substitutions
= { v: v*c for (v,c) in pairs }
943 return p
.subs(substitutions
)
947 def multiplication_table_from_matrix_basis(basis
):
949 At least three of the five simple Euclidean Jordan algebras have the
950 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
951 multiplication on the right is matrix multiplication. Given a basis
952 for the underlying matrix space, this function returns a
953 multiplication table (obtained by looping through the basis
954 elements) for an algebra of those matrices.
956 # In S^2, for example, we nominally have four coordinates even
957 # though the space is of dimension three only. The vector space V
958 # is supposed to hold the entire long vector, and the subspace W
959 # of V will be spanned by the vectors that arise from symmetric
960 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
961 field
= basis
[0].base_ring()
962 dimension
= basis
[0].nrows()
964 V
= VectorSpace(field
, dimension
**2)
965 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
967 mult_table
= [[W
.zero() for j
in xrange(n
)] for i
in xrange(n
)]
970 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
971 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
979 Embed the matrix ``M`` into a space of real matrices.
981 The matrix ``M`` can have entries in any field at the moment:
982 the real numbers, complex numbers, or quaternions. And although
983 they are not a field, we can probably support octonions at some
984 point, too. This function returns a real matrix that "acts like"
985 the original with respect to matrix multiplication; i.e.
987 real_embed(M*N) = real_embed(M)*real_embed(N)
990 raise NotImplementedError
996 The inverse of :meth:`real_embed`.
998 raise NotImplementedError
1002 def natural_inner_product(cls
,X
,Y
):
1003 Xu
= cls
.real_unembed(X
)
1004 Yu
= cls
.real_unembed(Y
)
1005 tr
= (Xu
*Yu
).trace()
1007 # It's real already.
1010 # Otherwise, try the thing that works for complex numbers; and
1011 # if that doesn't work, the thing that works for quaternions.
1013 return tr
.vector()[0] # real part, imag part is index 1
1014 except AttributeError:
1015 # A quaternions doesn't have a vector() method, but does
1016 # have coefficient_tuple() method that returns the
1017 # coefficients of 1, i, j, and k -- in that order.
1018 return tr
.coefficient_tuple()[0]
1021 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1025 The identity function, for embedding real matrices into real
1031 def real_unembed(M
):
1033 The identity function, for unembedding real matrices from real
1039 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
):
1041 The rank-n simple EJA consisting of real symmetric n-by-n
1042 matrices, the usual symmetric Jordan product, and the trace inner
1043 product. It has dimension `(n^2 + n)/2` over the reals.
1047 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1051 sage: J = RealSymmetricEJA(2)
1052 sage: e0, e1, e2 = J.gens()
1062 The dimension of this algebra is `(n^2 + n) / 2`::
1064 sage: set_random_seed()
1065 sage: n_max = RealSymmetricEJA._max_test_case_size()
1066 sage: n = ZZ.random_element(1, n_max)
1067 sage: J = RealSymmetricEJA(n)
1068 sage: J.dimension() == (n^2 + n)/2
1071 The Jordan multiplication is what we think it is::
1073 sage: set_random_seed()
1074 sage: J = RealSymmetricEJA.random_instance()
1075 sage: x,y = J.random_elements(2)
1076 sage: actual = (x*y).natural_representation()
1077 sage: X = x.natural_representation()
1078 sage: Y = y.natural_representation()
1079 sage: expected = (X*Y + Y*X)/2
1080 sage: actual == expected
1082 sage: J(expected) == x*y
1085 We can change the generator prefix::
1087 sage: RealSymmetricEJA(3, prefix='q').gens()
1088 (q0, q1, q2, q3, q4, q5)
1090 Our natural basis is normalized with respect to the natural inner
1091 product unless we specify otherwise::
1093 sage: set_random_seed()
1094 sage: J = RealSymmetricEJA.random_instance()
1095 sage: all( b.norm() == 1 for b in J.gens() )
1098 Since our natural basis is normalized with respect to the natural
1099 inner product, and since we know that this algebra is an EJA, any
1100 left-multiplication operator's matrix will be symmetric because
1101 natural->EJA basis representation is an isometry and within the EJA
1102 the operator is self-adjoint by the Jordan axiom::
1104 sage: set_random_seed()
1105 sage: x = RealSymmetricEJA.random_instance().random_element()
1106 sage: x.operator().matrix().is_symmetric()
1111 def _denormalized_basis(cls
, n
, field
):
1113 Return a basis for the space of real symmetric n-by-n matrices.
1117 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1121 sage: set_random_seed()
1122 sage: n = ZZ.random_element(1,5)
1123 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1124 sage: all( M.is_symmetric() for M in B)
1128 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1132 for j
in xrange(i
+1):
1133 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1137 Sij
= Eij
+ Eij
.transpose()
1143 def _max_test_case_size():
1144 return 4 # Dimension 10
1148 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1152 Embed the n-by-n complex matrix ``M`` into the space of real
1153 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1154 bi` to the block matrix ``[[a,b],[-b,a]]``.
1158 sage: from mjo.eja.eja_algebra import \
1159 ....: ComplexMatrixEuclideanJordanAlgebra
1163 sage: F = QuadraticField(-1, 'i')
1164 sage: x1 = F(4 - 2*i)
1165 sage: x2 = F(1 + 2*i)
1168 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1169 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1178 Embedding is a homomorphism (isomorphism, in fact)::
1180 sage: set_random_seed()
1181 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1182 sage: n = ZZ.random_element(n_max)
1183 sage: F = QuadraticField(-1, 'i')
1184 sage: X = random_matrix(F, n)
1185 sage: Y = random_matrix(F, n)
1186 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1187 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1188 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1195 raise ValueError("the matrix 'M' must be square")
1196 field
= M
.base_ring()
1199 a
= z
.vector()[0] # real part, I guess
1200 b
= z
.vector()[1] # imag part, I guess
1201 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1203 # We can drop the imaginaries here.
1204 return matrix
.block(field
.base_ring(), n
, blocks
)
1208 def real_unembed(M
):
1210 The inverse of _embed_complex_matrix().
1214 sage: from mjo.eja.eja_algebra import \
1215 ....: ComplexMatrixEuclideanJordanAlgebra
1219 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1220 ....: [-2, 1, -4, 3],
1221 ....: [ 9, 10, 11, 12],
1222 ....: [-10, 9, -12, 11] ])
1223 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1225 [ 10*i + 9 12*i + 11]
1229 Unembedding is the inverse of embedding::
1231 sage: set_random_seed()
1232 sage: F = QuadraticField(-1, 'i')
1233 sage: M = random_matrix(F, 3)
1234 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1235 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1241 raise ValueError("the matrix 'M' must be square")
1242 if not n
.mod(2).is_zero():
1243 raise ValueError("the matrix 'M' must be a complex embedding")
1245 field
= M
.base_ring() # This should already have sqrt2
1246 R
= PolynomialRing(field
, 'z')
1248 F
= NumberField(z
**2 + 1,'i', embedding
=CLF(-1).sqrt())
1251 # Go top-left to bottom-right (reading order), converting every
1252 # 2-by-2 block we see to a single complex element.
1254 for k
in xrange(n
/2):
1255 for j
in xrange(n
/2):
1256 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1257 if submat
[0,0] != submat
[1,1]:
1258 raise ValueError('bad on-diagonal submatrix')
1259 if submat
[0,1] != -submat
[1,0]:
1260 raise ValueError('bad off-diagonal submatrix')
1261 z
= submat
[0,0] + submat
[0,1]*i
1264 return matrix(F
, n
/2, elements
)
1268 def natural_inner_product(cls
,X
,Y
):
1270 Compute a natural inner product in this algebra directly from
1275 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1279 This gives the same answer as the slow, default method implemented
1280 in :class:`MatrixEuclideanJordanAlgebra`::
1282 sage: set_random_seed()
1283 sage: J = ComplexHermitianEJA.random_instance()
1284 sage: x,y = J.random_elements(2)
1285 sage: Xe = x.natural_representation()
1286 sage: Ye = y.natural_representation()
1287 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1288 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1289 sage: expected = (X*Y).trace().vector()[0]
1290 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1291 sage: actual == expected
1295 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1298 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1300 The rank-n simple EJA consisting of complex Hermitian n-by-n
1301 matrices over the real numbers, the usual symmetric Jordan product,
1302 and the real-part-of-trace inner product. It has dimension `n^2` over
1307 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1311 The dimension of this algebra is `n^2`::
1313 sage: set_random_seed()
1314 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1315 sage: n = ZZ.random_element(1, n_max)
1316 sage: J = ComplexHermitianEJA(n)
1317 sage: J.dimension() == n^2
1320 The Jordan multiplication is what we think it is::
1322 sage: set_random_seed()
1323 sage: J = ComplexHermitianEJA.random_instance()
1324 sage: x,y = J.random_elements(2)
1325 sage: actual = (x*y).natural_representation()
1326 sage: X = x.natural_representation()
1327 sage: Y = y.natural_representation()
1328 sage: expected = (X*Y + Y*X)/2
1329 sage: actual == expected
1331 sage: J(expected) == x*y
1334 We can change the generator prefix::
1336 sage: ComplexHermitianEJA(2, prefix='z').gens()
1339 Our natural basis is normalized with respect to the natural inner
1340 product unless we specify otherwise::
1342 sage: set_random_seed()
1343 sage: J = ComplexHermitianEJA.random_instance()
1344 sage: all( b.norm() == 1 for b in J.gens() )
1347 Since our natural basis is normalized with respect to the natural
1348 inner product, and since we know that this algebra is an EJA, any
1349 left-multiplication operator's matrix will be symmetric because
1350 natural->EJA basis representation is an isometry and within the EJA
1351 the operator is self-adjoint by the Jordan axiom::
1353 sage: set_random_seed()
1354 sage: x = ComplexHermitianEJA.random_instance().random_element()
1355 sage: x.operator().matrix().is_symmetric()
1360 def _denormalized_basis(cls
, n
, field
):
1362 Returns a basis for the space of complex Hermitian n-by-n matrices.
1364 Why do we embed these? Basically, because all of numerical linear
1365 algebra assumes that you're working with vectors consisting of `n`
1366 entries from a field and scalars from the same field. There's no way
1367 to tell SageMath that (for example) the vectors contain complex
1368 numbers, while the scalar field is real.
1372 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1376 sage: set_random_seed()
1377 sage: n = ZZ.random_element(1,5)
1378 sage: field = QuadraticField(2, 'sqrt2')
1379 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1380 sage: all( M.is_symmetric() for M in B)
1384 R
= PolynomialRing(field
, 'z')
1386 F
= NumberField(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1389 # This is like the symmetric case, but we need to be careful:
1391 # * We want conjugate-symmetry, not just symmetry.
1392 # * The diagonal will (as a result) be real.
1396 for j
in xrange(i
+1):
1397 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1399 Sij
= cls
.real_embed(Eij
)
1402 # The second one has a minus because it's conjugated.
1403 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1405 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1408 # Since we embedded these, we can drop back to the "field" that we
1409 # started with instead of the complex extension "F".
1410 return tuple( s
.change_ring(field
) for s
in S
)
1414 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1418 Embed the n-by-n quaternion matrix ``M`` into the space of real
1419 matrices of size 4n-by-4n by first sending each quaternion entry `z
1420 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1421 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1426 sage: from mjo.eja.eja_algebra import \
1427 ....: QuaternionMatrixEuclideanJordanAlgebra
1431 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1432 sage: i,j,k = Q.gens()
1433 sage: x = 1 + 2*i + 3*j + 4*k
1434 sage: M = matrix(Q, 1, [[x]])
1435 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1441 Embedding is a homomorphism (isomorphism, in fact)::
1443 sage: set_random_seed()
1444 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1445 sage: n = ZZ.random_element(n_max)
1446 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1447 sage: X = random_matrix(Q, n)
1448 sage: Y = random_matrix(Q, n)
1449 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1450 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1451 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1456 quaternions
= M
.base_ring()
1459 raise ValueError("the matrix 'M' must be square")
1461 F
= QuadraticField(-1, 'i')
1466 t
= z
.coefficient_tuple()
1471 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1472 [-c
+ d
*i
, a
- b
*i
]])
1473 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1474 blocks
.append(realM
)
1476 # We should have real entries by now, so use the realest field
1477 # we've got for the return value.
1478 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1483 def real_unembed(M
):
1485 The inverse of _embed_quaternion_matrix().
1489 sage: from mjo.eja.eja_algebra import \
1490 ....: QuaternionMatrixEuclideanJordanAlgebra
1494 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1495 ....: [-2, 1, -4, 3],
1496 ....: [-3, 4, 1, -2],
1497 ....: [-4, -3, 2, 1]])
1498 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1499 [1 + 2*i + 3*j + 4*k]
1503 Unembedding is the inverse of embedding::
1505 sage: set_random_seed()
1506 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1507 sage: M = random_matrix(Q, 3)
1508 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1509 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1515 raise ValueError("the matrix 'M' must be square")
1516 if not n
.mod(4).is_zero():
1517 raise ValueError("the matrix 'M' must be a complex embedding")
1519 # Use the base ring of the matrix to ensure that its entries can be
1520 # multiplied by elements of the quaternion algebra.
1521 field
= M
.base_ring()
1522 Q
= QuaternionAlgebra(field
,-1,-1)
1525 # Go top-left to bottom-right (reading order), converting every
1526 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1529 for l
in xrange(n
/4):
1530 for m
in xrange(n
/4):
1531 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1532 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1533 if submat
[0,0] != submat
[1,1].conjugate():
1534 raise ValueError('bad on-diagonal submatrix')
1535 if submat
[0,1] != -submat
[1,0].conjugate():
1536 raise ValueError('bad off-diagonal submatrix')
1537 z
= submat
[0,0].vector()[0] # real part
1538 z
+= submat
[0,0].vector()[1]*i
# imag part
1539 z
+= submat
[0,1].vector()[0]*j
# real part
1540 z
+= submat
[0,1].vector()[1]*k
# imag part
1543 return matrix(Q
, n
/4, elements
)
1547 def natural_inner_product(cls
,X
,Y
):
1549 Compute a natural inner product in this algebra directly from
1554 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1558 This gives the same answer as the slow, default method implemented
1559 in :class:`MatrixEuclideanJordanAlgebra`::
1561 sage: set_random_seed()
1562 sage: J = QuaternionHermitianEJA.random_instance()
1563 sage: x,y = J.random_elements(2)
1564 sage: Xe = x.natural_representation()
1565 sage: Ye = y.natural_representation()
1566 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1567 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1568 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1569 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1570 sage: actual == expected
1574 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1577 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1579 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1580 matrices, the usual symmetric Jordan product, and the
1581 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1586 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1590 The dimension of this algebra is `2*n^2 - n`::
1592 sage: set_random_seed()
1593 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1594 sage: n = ZZ.random_element(1, n_max)
1595 sage: J = QuaternionHermitianEJA(n)
1596 sage: J.dimension() == 2*(n^2) - n
1599 The Jordan multiplication is what we think it is::
1601 sage: set_random_seed()
1602 sage: J = QuaternionHermitianEJA.random_instance()
1603 sage: x,y = J.random_elements(2)
1604 sage: actual = (x*y).natural_representation()
1605 sage: X = x.natural_representation()
1606 sage: Y = y.natural_representation()
1607 sage: expected = (X*Y + Y*X)/2
1608 sage: actual == expected
1610 sage: J(expected) == x*y
1613 We can change the generator prefix::
1615 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1616 (a0, a1, a2, a3, a4, a5)
1618 Our natural basis is normalized with respect to the natural inner
1619 product unless we specify otherwise::
1621 sage: set_random_seed()
1622 sage: J = QuaternionHermitianEJA.random_instance()
1623 sage: all( b.norm() == 1 for b in J.gens() )
1626 Since our natural basis is normalized with respect to the natural
1627 inner product, and since we know that this algebra is an EJA, any
1628 left-multiplication operator's matrix will be symmetric because
1629 natural->EJA basis representation is an isometry and within the EJA
1630 the operator is self-adjoint by the Jordan axiom::
1632 sage: set_random_seed()
1633 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1634 sage: x.operator().matrix().is_symmetric()
1639 def _denormalized_basis(cls
, n
, field
):
1641 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1643 Why do we embed these? Basically, because all of numerical
1644 linear algebra assumes that you're working with vectors consisting
1645 of `n` entries from a field and scalars from the same field. There's
1646 no way to tell SageMath that (for example) the vectors contain
1647 complex numbers, while the scalar field is real.
1651 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1655 sage: set_random_seed()
1656 sage: n = ZZ.random_element(1,5)
1657 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1658 sage: all( M.is_symmetric() for M in B )
1662 Q
= QuaternionAlgebra(QQ
,-1,-1)
1665 # This is like the symmetric case, but we need to be careful:
1667 # * We want conjugate-symmetry, not just symmetry.
1668 # * The diagonal will (as a result) be real.
1672 for j
in xrange(i
+1):
1673 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1675 Sij
= cls
.real_embed(Eij
)
1678 # The second, third, and fourth ones have a minus
1679 # because they're conjugated.
1680 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1682 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1684 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1686 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1692 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1694 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1695 with the usual inner product and jordan product ``x*y =
1696 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1701 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1705 This multiplication table can be verified by hand::
1707 sage: J = JordanSpinEJA(4)
1708 sage: e0,e1,e2,e3 = J.gens()
1724 We can change the generator prefix::
1726 sage: JordanSpinEJA(2, prefix='B').gens()
1730 def __init__(self
, n
, field
=QQ
, **kwargs
):
1731 V
= VectorSpace(field
, n
)
1732 mult_table
= [[V
.zero() for j
in xrange(n
)] for i
in xrange(n
)]
1742 z0
= x
.inner_product(y
)
1743 zbar
= y0
*xbar
+ x0
*ybar
1744 z
= V([z0
] + zbar
.list())
1745 mult_table
[i
][j
] = z
1747 # The rank of the spin algebra is two, unless we're in a
1748 # one-dimensional ambient space (because the rank is bounded by
1749 # the ambient dimension).
1750 fdeja
= super(JordanSpinEJA
, self
)
1751 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
1753 def inner_product(self
, x
, y
):
1755 Faster to reimplement than to use natural representations.
1759 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1763 Ensure that this is the usual inner product for the algebras
1766 sage: set_random_seed()
1767 sage: J = JordanSpinEJA.random_instance()
1768 sage: x,y = J.random_elements(2)
1769 sage: X = x.natural_representation()
1770 sage: Y = y.natural_representation()
1771 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1775 return x
.to_vector().inner_product(y
.to_vector())